DC motor modeling and control loop design
Motor description
System modeling
Electric equation:
From kirchhoff's voltage law :
Mechanical equation:
By Newton's law:
Electro-mechanical coupling:
The back EMF is proportional to speed :
The motor torque is proportional to current :
Motor Bloc Diagram
Motor control : cascaded strategy
The control synthesis is inspired by Permanent Magnet Synchronous Motor control synthesis based on cascaded control synthesis. Due to frequency separation the control can be divided into two control loops. The inner loop control the electrical dynamic while the outer loop treats the mechanical dynamic. Generally the the electrical dynamics is neglected and the mechanical dynamics is considered only. However in the case where motor resistance is low, this strategy can damage the motor.
Electrical dynamics control
With the assumption that the mechanical dynamic is slower the the electrical one, one has :
Feedback control with integral action
The electrical dynamics is given by
From the figure one gets :
The system dynamics can be described by
with the control
The closed loop system leads to :
The closed loop dynamics depends on the eigenvalues of the matrix :
One has
It leads to a characteristic equation
to be identified with the classical second order characteristic equation
Mechanical dynamics control
Assuming the electrical control has been correctly synthesized with respect to frequency separation principle, which means that the closed loop electrical dynamics is faster than the mechanical desired dynamics, then the mechanical dynamics control synthesis can be designed without considering the closed loop electrical system. The control scheme can be simplified as :
The mechanical dynamics is
with the control
By analogy, it leads to a characteristic equation
to be identified with the classical second order characteristic equation
References
(Chiasson2005) Chiasson, J.-N. (2005). Modeling and High-Performance Control of Electric Machines (IEEE Press).
(Ogata2010) Ogata, K. (2010). Modern Control Engineering. Prentice Hall.
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